Spinning magnetic fields
Jovan Djuric
Department of Electrical Engineering and Computer Science, The University of New Mexico,
Albuquerque, New Mexico 87131
(Published in Journal of Applied Physics, vol. 46, No. 2, February 1975)
Brief Summary
A possible electrical charge model based on the spinning time invariant point magnetic dipole within the
framework of classical physics is outlined, as suggested by the admissible circular trajectory of a test charge
around the magnetic dipole in its equatorial plane. The model depends on the moving force line hypothesis
which has been claimed to have been disproved. The controversy about that issue at the turn of this century
is reanalyzed. The mathematical model of the generalized homopolar generator is presented, which is fully
in agreement with all experimental facts concerning the homopolar generator, except for a single result
by Pegram which appears to be an error and which violates the Faraday law. It is shown that the moving
force line hypothesis was apparently not disproved by Kennard as claimed. Thus the intriguing model of
electrical charge is possible. Further experimental research with all kinds of spinning magnets appears to
be justified and desirable in view of the potentially high significance of the research line initiated by Jehle
with the quantum mechanical model of the charge in terms of spinning flux loop forms.
INTRODUCTION
The movement of the lines of force of a magnetic field whose source is rotating represents a very
old puzzle dating back to Faraday’s discovery of the homo­polar generator or the Faraday disk,
as it is also re­ferred to in the literature.
The fact is incontestable that the lines of force of a magnetic field do move with the movement of the source of that magnetic field, as long as ∂B/∂t is not equal to zero in the coordinate system in which the movement of the source is observed and in which the field B is de­
fined. This fact can be easily proved experimentally.
However, if there is the axial symmetry of a steady magnetic field and its source, and if that
source is ro­tated around that axis of symmetry with the constant angular velocity, it may appear
as though the lines of force do not rotate with the rotation of the source, and that such a rotation
is not detectable electromagnetical­ly since ∂B/∂t ≡ 0 in such a case. This problem arose directly in connection with the homopolar generator. The numerous attempts to resolve that problem
led to a controversy, which is described in several papers,1-5 at the beginning of this century. The
additional refer­ences to the very voluminous and often very contradic­tory literature on the homopolar generator can be found in those papers.
A quantum-mechanical electrical charge model based on the spinning flux loop forms was
recently proposed and extensively analyzed by Jehle.6,7 Independently, the present author proposed an electrical charge model based on the spinning point magnetic dipole strictly within
the framework of classical physics. It is in con­nection with that model that the old puzzle of the
moving or the nonmoving force line hypothesis surfaced once again.
35
The objectives of this paper are to indicate the possi­bility of a new model of the electrical
charge and to present a new mathematical model of the so-called gen­eralized homopolar generator in order to discuss the old but apparently still unresolved problem of the mov­ing force
line hypothesis or theory.
POINT MAGNETIC DIPOLE
Consider a point magnetic dipole located at the coor­dinate origin, whose magnetic induction
density field is given by
,
(1)
where the notation is conventional. The rationalized MKS system of units is utilized. The magnetic dipole moment m is assumed to be constant and directed along the z axis.
Let a point charge q of mass m0 move in the field with velocity v, whose magnitude is assumed to be much smaller than c so that the variation of mass with velocity can be neglected.
The equation of motion is
,
(2)
where B is given by Eq. (1).
This equation of motion permits the circular trajec­tory to be a possible solution if for a constant w the velocity vector v is in the equatorial plane,
,
(3)
so that
.
(4)
The possible circular trajectory of a point electrical charge around a magnetic dipole as
specified above is apparently observed as ring currents associated with the geomagnetic storms.
The right-hand side of Eq. (4) can be written
,
(5)
with
.
(6)
It is clear from the above that the observer can inter­pret Eq. (5) as though the point charge
q revolves around another point charge - Q in the equatorial plane, i. e., under the conditions
specified the point magnetic dipole appears as a point charge and is absolutely indistinguish­
able from a point charge.
36
This simple but admissible solution of the rather very complex dynamic problem raises the
question: What hap­pens if it is assumed that the magnetic dipole itself ro­tates with the same
constant angular velocity ω ?
First of all, the argument that the rotation of a point magnetic dipole, i.e., a point structure,
is physically meaningless must be immediately dismissed. The point magnetic dipole is a mathematical category which is used to describe and/or approximate the observed phenomena in nature. It stands for some real objects or particles possessing the magnetic property. It can be rotated by definition.
The answer to the above question appears to be a puz­zle. Namely, the magnetic field given
by the expression (1) represents, in view of the assumed axial symmetry, a continuum of lines
which are not discernible mathema­tically in any way whatsoever when the transformation is
made from one system to the other having the com­mon coordinate origin and the common z
axis and ro­tating with respect to each other around the axis of sym­metry, i. e., the z axis.
Nevertheless, it appears that there are two logically possible answers to the above question.
(i) There is no change of the circular trajectory of the test point electrical charge due to the
rotation of the magnetic dipole, i. e., the rotation of the magnetic dipole around its axis of symmetry is of no consequence whatsoever and is not detectable by any electromagnetic means. In
other words, the magnetic field is not car­ried by the rotating magnetic dipole. This is the nonmoving force line hypothesis or theory, or the N hypoth­esis for short. It consistently implies
that the magnetic field of that axially symmetrical magnetic dipole is an abstraction rather than
a physical reality.
(ii) The magnetic dipole carries its own magnetic field with the rotation and, as a consequence, the rotat­ing test charge no longer is moving with respect to the magnetic field, since the
magnetic dipole and the test charge rotate together by assumption. In such a case the centrifugal acceleration on the rotating test charge is observed only by a stationary observer and must
be balanced by some other force. This second alternative is referred to as the moving force line
hypothesis or theory, or the M hypothesis for short.
Purely logically, it may be argued that there is an in­finite number of possibilities between
those two limiting alternatives. The problem itself is sufficiently puzzling with only two hypotheses, so only those two hypotheses will be analyzed.
POSSIBLE ELECTRICAL CHARGE MODEL
The M hypothesis admits the following electrical charge model. If the magnetic dipole spins with
the constant angular velocity ω=ωaz , then the stationary test charge in the stationary coordinate
system S experi­ences the electric field by assumption
,
(7)
37
with the minus sign as appropriate and B given by Eq. (1), since B due to the assumed absence
of the ϕ depen­dence remains the same in all systems having the com­mon z axis and the common coordinate origin.
Introducing Eq. (1) into Eq. (7), the electric field E becomes
,
(8)
where the notation is conventional for the spherical coordinate system.
It is easy to show that
,
(9)
with
.
(10)
It is clear from the above expressions that the elec­tric field observed by the stationary observer subject to the M hypothesis is strictly conservative by definition, although it is dynamic and
magnetic according to its origin, It is not the Coulomb-type field, except in the equatorial plane.
Another puzzling or interesting fea­ture of the electric field given by the expression (8) is that its
divergence is not equal to zero anywhere in the space, which Sommerfeld refers to as “an interesting mathematical difficulty” (cf. Ref. 8, p. 363). In the equatorial plane the electric field may be
considered as due to the electrical charge Q given by Eq. (6), with the positive sign. Thus, subject
to the validity of the M hy­pothesis, the spinning point magnetic dipole appears as a charge + Q as
far as a stationary test charge in the equatorial plane is concerned. This is in full symmetry with
the fact that for a test charge circling around a stationary point magnetic dipole in the equatorial plane the magnetic dipole appears as a charge — Q. Actually, the observer attached to that revolving test charge must observe that the magnetic dipole is circling around the test charge, and
simultaneously the magnetic dipole is spinning with the synchronous angular velocity in the negative direction, hence there is the change of sign. There is a perfect symmetry between those two
obser­vations if the validity of the M hypothesis is assumed, and this is a strong argument in favor of the M hypoth­esis. The change of sign is the familiar feature from the homopolar generator.
It should be noted that the force exerted upon the circling point electrical charge by the
magnetic dipole is presumably well understood, i. e., the Lorentz force given by Eq, (2). However, under the conditions speci­fied, there is no force whatsoever upon the magnetic dipole due
to the circulating charge, and Newton’s third law of equality of action and reaction appears to
be vio­lated unless the spinning magnetic dipole spinning in the negative mathematical direction relative to the electri­cal charge is equivalent to the electrical charge. If and only if the spinning magnetic dipole is equivalent to the electrical charge, Newton’s third law is not violated in
this case. This argument appears to strongly favor the M hypothesis. If, on the other hand, the
spinning mag­netic dipole is not equivalent to the electrical charge, then this case of the interaction between the magnetic dipole and the electrical charge represents not only the breakdown
of Newton’s third law, but also the break­down of the classical electromagnetic field theory.
38
In order to define the equivalent charge associated with the electric field [Eq. (8)], the calculation of the flux of that electric field over any concentric sphere having its center at the coordinate origin may be used, which yields
.
(11)
If m is assumed to be the Bohr magneton 9.27×10-24 Am2, and if Qeq is equated to the quantum (electron) charge of 1.6×10-19 C, then Eq. (11) yields for the mag­nitude of the angular velocity
.
(12)
This is the possible charge model which exhibits the 6 dependence (axial dependence). For
a macroscopic charge distribution involving a very large number of such model charges with different random orientations of axes, the averaging over-all possible orientations may lead to the
Coulomb field. The axial dependence may offer the possibility to account for the very intrigu­
ing “holes” within the framework of classical physics.
It should be mentioned that Jehle6,7 defines his quan­tum-mechanical charge model by averaging over all possible orientations of the axis of the model itself. Jehle’s flux loop form corresponds fully to the point magnetic dipole. The averaging used by Jehle is inad­missible within
classical physics by definition.
The angular velocity [Eq. (12)] yields for the maxi­mum circular velocity of the electron model a value which is well below c, if the classical electron radius of 2.817×10-15 m is used.
There is a puzzling feature of this possible electrical charge model. Namely, the electric field
[Eq. (8)] is magnetic according to its origin. As such, it cannot presumably be screened by a nonferromagnetic metallic conductor. But a true electrostatic field can be screened by metallic conductors, which is the proof of the Cou­lomb law. A large number of the above-described charge models with random axial orientations unques­tionably yields the Coulomb-type field which must be
screenable, but it remains obscure how and why. Never­theless, Jehle’s model, as well as the possible
charge model outlined above, represent intriguing possibilities which should be explored in spite
of some difficulties. This conclusion means that it is worthwhile to reex-amine the problem of the
M or the N hypothesis. Name­ly, Kennard3,4 claimed that the M hypothesis was ex­perimentally disproved, while Barnett1,2 maintained that the M hypothesis was not at all disproved. The contro­versy
died in view of the preoccupation of the physicists of that era with Bohr’s revolutionary ideas, and
Ken-nard’s claim was accepted because the physicists were tired of that problem. It will be shown
in this paper that all experimental results are fully consistent with either the M or the N hypothesis.
HOMOPOLAR GENERATOR
The homopolar generator in its classical form as dis­covered and described by Faraday consists of an
axially symmetrical permanent magnet, which is rotated around its axis of symmetry, while a stationary metallic wire is connected to two brushes sliding against two different points of the rotating magnet (see Fig. 1). The equatori­al point and the end point of the magnet are the typical points
39
for the brushes to be placed in order to obtain the largest possible electrical current, which is observed in the described circuit. If the magnet is stopped while the wire is rotated around the same
axis with the same constant angular velocity, but in the opposite direction, the same steady electrical
current is observed flowing through that circuit. It is assumed that the permanent magnet is made of
an electrically conducting material, thus forming the part of the closed electrical circuit. If a steady
electromagnet is used instead of the perma­nent magnet, then an appropriate coaxial conductor, in­
sulated from the windings of that electromagnet, must cover the electromagnet and rotate with it.
The question now arises: What is the explanation of the phenomenon and which part of the circuit is the seat of the induced electromotive force (EMF), the rotating magnet or the stationary wire?
In order to answer that question, the moving and the nonmoving force line theories or hypotheses have been put forward. The M hypothesis assumes that the ro­tating magnet carries
its own magnetic induction den­sity field, which rotates with the magnet together. The M theory leads obviously to the conclusion that the sta­tionary wire is the seat of the induced EMF and
not the rotating magnet. Of course, if the magnet is stationary while the wire rotates the problem disappears, since in that case the wire is obviously the seat of the induced EMF. The second hypothesis, i. e., the N hypothesis, assumes that the rotating magnet does not carry its own
magnetic field during the rotation because of the axial symmetry. The N hypothesis leads to the
conclusion that it is the rotating magnet which is the seat of the induced EMF, i. e., the rotating
magnet cuts the lines of its own but standing magnetic field.
The above question is somewhat oversimplified, since the classical form of the homopolar
generator is also oversimplified. Namely, in the case of the steady elec­tromagnet instead of the
permanent magnet, it has been already mentioned that a separate coaxial conductor must be
present, which must be electrically insulated from the windings of that electromagnet. The coaxial conductor may be a cylinder, a disk, a frustum (cone), or any other coaxial form. The disk
with the two brush­es attached to two points with different radii is a con­figuration which is referred to as the Faraday disk, and is quite simple and straightforward. The disk can be easily arranged to rotate independently from the magnet around the same axis of symmetry, but with a
different constant angular velocity. As a matter of fact, from the electrical point of view the disk
is an axially symmetri­cal structure which is fully defined by the impedance Z1 = Z1(s) offered by
the disk at the points of the two brushes, denoted as brushes A and B. The variable s is the familiar complex frequency of the Laplace transform.
FIG. 1. Classical homopolar generator.
40
FIG. 2. Generalized homopolar generator. Note, the imped­ances Z1 and Z2
may be located inside the magnet.
It should be mentioned that in the experimental mea­surements by Barnett,1,2 and Kennard,3,4
coaxial ca-pacitive structures within the almost uniform coaxial magnetic field were used as parts
of what was essen­tially a homopolar generator. Kennard’s approach, re­ported in Ref. 4, with a
rotating coaxial capacitor and a stationary electrometer, i. e., stationary capacitor, leads in particular to the concept of the generalized homopolar generator which is defined here as a natural outgrowth of the experimental setups used by Barnett1 and Kennard.4
The impedance Z1 is almost purely real (resistive) in the case of a disk. Experiments by Kennard and other physicists (viz., Ref. 2, p. 325) unmistakably show that a coaxial capacitor rotating in a magnetic field whose direction coincides with the axis of rotation is a seat of an induced
EMF, since such a rotating capacitor can charge a stationary capacitor, i.e., an electrometer. This
is a very important experimental result since for the general case of the homopolar generator the
disk can be substituted by a coaxial capacitor of the appro­priate length, in which case Z1 is approximately equal to l/sC1, where C1 is the capacitance of that coaxial capacitor, hence neglecting the small resistances and inductances of the leads and the brushes as well as the large but
inevitable leakage resistances of the insula­tors carrying the armatures of that coaxial capacitor.
4
Thus, the generalized homopolar generator which is very suitable for the analysis is formed
as follows and is sketched in Fig. 2. Let the axis of symmetry of the steady magnet be the z axis.
The steady magnet ro­tates around the z axis with the constant angular velocity ωMaz . Let the
axially symmetrical structure char­acterized electrically by and called the impedance Z1 rotate
around the z axis with the constant angular ve­locity ωM= ω1az . The impedance Z1 is connected
by two brushes A and B, which may be stationary or rotating, to another impedance Z2 (instead
of a simple wire and an ammeter in the classical arrangement of the homo-polar generator) in
series with a switch SW, which is controlled remotely from the stationary (laboratory) coordinate system S. The switch SW is assumed to be an idealized one, with the impedance between
41
its two terminals equal to infinity when the switch is open and equal to zero when the switch is
closed. The practical switches approximate those conditions only within cer­tain frequency bands.
Let the impedance Z2 , which includes all connecting-wires, and the series switch SW rotate
around the z axis with the constant angular velocity ω2= ω2az . Since by definition impedances
Z1 and Z2 must be time-invariant functions of the complex frequency s only, it is important to
note that only one of the two impe­dances must be an axially symmetrical structure in order to
provide circular sliding surfaces for the two brushes A and B. Since it has been already assumed
that Z1 is an axially symmetrical structure, impedance Z2 need not be axially symmetrical. The
switch SW is necessary in order to define the zero time, and also to disconnect impedances Z1
and Z2 at any chosen instant of time. The movement of the moving parts of switch SW is assumed to be strictly radial, so that such a movement does not interfere with the induced electric
field due to the circular (rotational) movement of branch Z2 + SW of the circuit of the homopolar generator. This is a closed circuit by definition, even if impedance Z1 is that of a coaxial capacitor, which represents a short before it is charged during the transient period.
The above described configuration of the three rotat­ing structures, each rotating with its own
constant angu­lar velocity, represents a generalized homopolar gene­rator, In order to avoid the
thermal and the contact po­tential effects, it is assumed that all metallic parts of the circuit of the
generalized homopolar generator are made of the same metal, and are at the same tempera­ture.
The magnet can be connected to a single refer­ence point of the circuit of the homopolar generator, the ground point, without disturbing the operation of the generalized homopolar generator.
In the forthcoming analysis of the generalized homo-polar generator, the thermal effects,
the electromag­netic radiation effects, the thermal noise, the contact potential effects, the inertial effects, and the elastic or plastic deformation effects are assumed to be negli­gibly small and
are neglected.
ANALYSIS OF GENERALIZED HOMOPOLAR GENERATOR
Let us assume that the three angular velocities of the described generalized homopolar generator have reached the steady states. An instrument, a storage, i.e., memory oscilloscope for example, assumed to be insensitive to acceleration and with a very high input impedance is attached
to impedance Z2 in order to ob­serve and record the flow of the electric current through the circuit of the homopolar generator. The instrument is rotating together with the impedance Z2 , It
is also possible to form special brushes Bʹ and Bʹʹ instead of a single one, so that the stationary
observer can con­nect an instrument (ammeter) in series with the circuit of the homopolar generator without disturbing the opera­tion of the circuit in any significant way. All three angu­lar
velocities, the switch SW, and the instrument are monitored and controlled from the laboratory
(station­ary) coordinate system S.
Let us close the switch SW at one instant of time, thus defining the zero time for the experiment. The electric current i = i(t) is presumably observed to flow through the circuit of the homopolar generator, due to the in­duced EMF u = u(t) within that circuit.
42
Let I=I(s) = L{i} denote the Laplace transform of the current flowing through that circuit.
Let U= U(s) = L{u} be the Laplace transform of the induced EMF. The no­tations are conventional for the theory of the Laplace transforms. The zero initial conditions of all elements of the circuit of the homopolar generator are assumed. Thus,
.
(13)
represents the Laplace transform of the conventional Kirchhoff voltage law applied to the closed
loop formed by impedances Z1 and Z2 of the homopolar generator. Note that u = L-1{U} is the
total induced EMF within that circuit. In order to calculate that induced EMF, the in­duced electric fields acting within the metallic and/or dielectric parts of those two impedances (branches) must be obtained.
For the axially symmetrical permanent magnet of the homopolar generator,
.
No free true (already separated) electric charges are assumed anywhere in the space of the homopolar generator. Thus, the electric field acting upon the charges within impedances Z1 and
Z2 is the well-known motional term
, which can be directly derived from the general Far-
aday law of the electromagnetic induction in the integral form
be valid without any restrictions whatsoever.
, which is assumed to
In the expression
,
(14)
v is the velocity of the moving charge relative to the magnetic induction density field
relative to the coordinate system in which that field
tion field
, i. e.,
is defined. But the magnetic induc-
of the axially sym­metrical permanent magnet is a peculiar one. It de­pends on
R= + (x + y ) and z only, and not on ϕ by definition. Thus, the lines of force of that field form
a continuum which is not discernible mathematically from one coordinate system to the other,
if those systems have the common coordinate origin and the common z axis, which is the axis
of symmetry of that magnet by assumption. So the question does arise: Which velocity must be
used in calculations if the magnet itself ro­tates? It is in that respect that the moving and the nonmoving force line hypotheses have been proposed.
2
2 1/2
The axially symmetrical permanent magnet (or steady electromagnet) creates a steady magnetic induction den­sity field which can be described and/or approximated by using either
the scalar potential or the vector poten­tial. Both methods are fully equivalent as long as no true
electrical current is encircled in the calculation of the EMF, which is true for the case analyzed
here in this paper. The scalar potential method is chosen here since it is somewhat simpler for
the mathemati­cal manipulations. The geometrical center of the mag­net is assumed to coincide
with the origin of the cylin­drical coordinate system of reference S. The cylindri­cal coordinate
system is the most appropriate one in this case, in view of the axial symmetry of the problem.
43
Let the magnetic induction density field
be rep­resented by
,
(15)
where Ψ is the magnetic scalar potential, which is by assumption a well-behaved function of R
and z only, i. e., the ϕ dependence is absent by assumption. The notation is conventional.
Since no magnetic monopoles are to be assumed or utilized in this paper,
by definition. This limits the applicability of expression (15) for
to the region outside the permanent magnet, but includes all points on the surface of the permanent magnet which suffices for
the analysis here in this paper. On the other hand, expression (15) can be used in the case of the
solenoid-al electromagnet inside as well as outside the solenoid as long as during the calculation of the EMF the closed path of integration never encircles the true electric current. All the
above conditions are met in the cal­culations of the induced EMF in the case of the general­ized
homopolar generator analyzed in this paper.
In view of
, the scalar magnetic potential
which satisfies the Laplace equation
is a harmonic function
(16)
Let us assume that the velocity v of the electrical charge rotating within its branch (impedance) is
,
(17)
where ω is the algebraic value of the constant angular velocity of the rotating charge, to be determined in terms of ωM , ω1 and ω2 depending on the hypothesis assumed, and for which branch
of the circuit the velo­city Eq. (17) is to be applied.
Introducing Eqs. (15) and (17) into expression (14),
becomes
, after some simple vector algebra,
.
(18)
This electric field can also be expressed in the form
,
(19)
where W= W(R,z) is an appropriate scalar function. Equating the corresponding components
from Eqs. (18) and (19), the following equations are obtained:
,
and
44
(20)
Cancelling ω in the above equations and taking the par­tial derivative of Eq. (20) with respect
to z and the partial derivative of Eq. (21) with respect to R, the fol­lowing expression is obtained
.
(22)
This expression is the identity in view of the Laplace equation [Eq. (16)] for Ψ . Of course
the interchange of the order of the partial differentiation of Ψ(R, z) with respect to R and z is
assumed to be permissible, which is true under some very broad conditions invariably met in
all physical problems.
Thus, it is proven that E given by Eq. (18), i. e., Eq. (19), is indeed a conservative electric
field under the conditions specified. However, this electric field is not equivalent to a true electrostatic field since this electric field is motional and magnetic according to its origin. It cannot be screened by enveloping closed non-ferromagnetic conductors since it is not a Coulombtype electrostatic field, which is screenable by nonferro-magnetic conductors according to the
experimental evidence. The actual evaluation of W(R, z) requires the detailed specification of
the geometrical configuration and either the magnetization density vector or the elec­tric current density vector.
CALCULATION OF EMF
The calculation of the induced EMF within the circuit of the generalized homopolar generator
will be done separately for both of the logically possible hypotheses (or theories): The M (moving force line) hypothesis (Case 1) and the N (nonmoving force line) hypothesis (Case 2).
Case 1
The M hypothesis is assumed to be valid in this case. Thus, the magnetic induction density field
B of the axial­ly symmetrical steady magnet of the generalized homo­polar generator rotates together with the magnet with the constant angular velocity ωM , Consequently, branch Z1 of the
homopolar generator rotates relative to the magnetic induction density field B with the constant
angular velocity ω1 – ωM = (ω1 – ωM) az , which is obvious in view of the description and the definition of the gen­eralized homopolar generator. Thus, the induced elec­tric field acting upon the
electrical charges within branch Z1, due to their rotation is obtained by substitut­ing w in the expression (19) with ω1 – ωM , which yields
,
(23)
where the index 1 in El m denotes that the induced elec­tric field is applicable for branch Z1; while
the second subscript (index) m denotes that the induced electric-field is applicable with the M
hypothesis assumed to be valid.
In view of the M hypothesis, branch Z2 rotates with respect to the rotating magnetic induction density field B with the constant angular velocity ω2 – ωM = (ω2 – ωM) az . Thus, using
45
the expression (19), the induced electric field acting upon the rotating electrical charges within
branch Z2 of the circuit of the generalized homopolar generator is given by
,
(24)
where the subscript m has the same meaning as in Eq. (23).
Note that expressions (23) and (24) are valid quite generally for all algebraic values of ωM ,
ω1 and ω2 , although in the derivations of the relative angular ve­locities it is most natural to imagine that ωM is smaller than both ω1 and ω2 , while all three values are positive.
The induced EMF within the circuit is by definition the strictly closed-loop integral of the
electric field around the closed loop. If Em denotes formally that elec­tric field, which within the
appropriate branches of the circuit is defined by Eqs. (23) or (24), then um = um(t), i.e., the induced EMF in this Case 1 is given by
(25)
,
where WA and WB denote the values of the potential function W(R, z) at the brush points A and
B, respec­tively. Note that the circulation in the above integration around the closed loop has been
performed strictly in the positive mathematical direction.
It is clear that the induced EMF um as given by Eq. (25) is independent of the angular velocity ωM of the ro­tating magnet.
Case 2
The N hypothesis is assumed to be valid in this case. In view of this hypothesis, the magnetic induction den­sity field B of the axially symmetrical steady magnet of the generalized homopolar
generator does not rotate with the rotating magnet by assumption.
Consequently, branch Z1 of the generalized homo-polar generator rotates relative to the magnetic induc­tion density field B with the constant angular velocity ω1 = ω1 az by definition. Thus,
the induced electric field acting upon the electrical charges rotating with branch Z1 is given by
,
(26)
where the subscript 1 in E1n denotes that the induced electric field is applicable for branch Z1
while the second subscript n denotes that that induced electric field is applicable with the N hypothesis assumed to be valid.
For branch Z2 , which rotates with respect to the mag­netic induction density field B with
the constant angular velocity ω2 = ω2 az by assumption, the induced electric field is
,
where the subscript n has the same meaning as in Eq. (26).
46
(27)
The induced EMF un = un(t) is given by
(28)
The circulation in the above integration around the closed loop has been performed strictly in
the positive mathematical direction, i.e., exactly as in Eq. (25).
The comparison of Eqs, (25) and (28) yields
.
(29)
Since the above analysis is quite general and applica­ble to the transient as well as to the
steady-state mea­surements of the current which flows through the cir­cuit of the generalized
homopolar generator, which must be considered as closed by definition during the transient
periods, the conclusion is reached from the above result that no experiment with the generalized homopolar generator or its classical form can resolve the puzzle, which one of the two logically possible hy­potheses is correct, the moving force line hypothesis or the nonmoving force line
hypothesis.
The above conclusion has been obtained using the in­tegral approach of the circuit theory, which is a branch of the classical electromagnetic field theory. Conse­quently, the differential approach must yield the same result. However, it is obvious that the differential approach
to the solution of this problem is plagued with many difficulties, such as the boundary conditions, the geometrical configurations of branches Z1 and Z2 , the switch SW, the brushes, etc.
The above integral solu­tion has resolved only the problem of the flow of the electrical current
through the circuit of the generalized homopolar generator, and, consequently, it is possible
to calculate the final values of the voltages across im­pedances Z1 and Z2 , provided those impedances are known over the entire frequency spectrum with a rea­sonable accuracy. However, this integral solution can­not and does not give the field distribution within the generalized
homopolar generator after the transient period is over.
The preceding analysis clearly shows that the rela­tive rotation of the branches of the circuit of the gene­ralized homopolar generator is essential for the flow of the electrical current
through the circuit of that gen­erator. Using this mathematical model of the general­ized homopolar generator, which is fully consistent with all experimental results, except for a single
result by Pegram,5 it will be shown that the claim by Kennard, 3,4 that the M hypothesis was
disproved, ap­pears to be definitely erroneous. Moreover, if the M hypothesis is to be assumed
to have been disproved, then that may be due only to Pegram,5 who never claimed that, and
whose result appears to violate the Faraday law of induction.
47
BRIEF HISTORICAL REVIEW
A brief historical review is included here in order to appreciate the anguish of many physicists
who have attempted to find the solution to the probelm of whether the M or the N hypothesis
is correct, contributing in the process to the voluminous and contradictory litera­ture on the
subject of the homopolar generator (see Refs. 1-5).
The question about the seat of the EMF of the homo-polar generator in its classical form,
which problem is directly connected with the M and N hypothesis, was raised originally by
the discoverer of the homopolar generator, i. e., Faraday himself. Faraday expressed vague
belief that the seat of the EMF was the rotating magnet, which is an oversimplification of the
problem. Thus, Faraday favored the N hypothesis, which belief probably influenced the debate concerning the issue. Faraday experimented only with the classical form of the homopolar generator.
In order to experimentally resolve the puzzle of the M or the N hypothesis, it occurred
to Barnett,1 p. 324, in 1902 (cf. also Fig. 2) “that the problem could be solved by the following
method. A cylindrical condenser (capacitor) is placed in an approximately uniform mag­netic
field parallel to its axis, and the magnets pro­ducing the field are rotated while the condenser is
short circuited by a wire at rest like itself. While the mag­nets are still rotating, the connection
between the arma­tures of the condenser is broken; and the condenser is tested for charge after the field is annulled, or the rotation stopped, or the condenser removed. It was argued that
if the lines of induction moved with the mag­nets the condenser would receive charges which
could be computed, and that it would remain uncharged in case the lines remained fixed and
the magnets moved through them. It was found later that a somewhat simi­lar idea, discussed
below, had previously occurred to Preston,9 and in 1908 Waring10 proposed an experiment essentially the same in principle as that which occurred to me. As shown below, however, our
reasoning was erroneous. Since what precedes was written there has appeared an experimental paper on the subject by Ken­nard3 which also is based on incorrect theory.”
Barnett argued in his paper1 that the basic premise of his experiment as well as that of Kennard was wrong. Barnett used rather weak arguments and even some er­roneous arguments.
Barnett concluded1 that the experi­ment could not prove or disprove the M hypothesis. Barnett’s
experimental results were unmistakable and showed no charge in the condenser (capacitor).
In his second paper2 and elsewhere, Barnett con­tinued the apparently futile controversy with Kennard, but was obviously unsuccessful. Namely, Barnett was upstaged by Kennard,3
who only a few months before the publication of the thorough paper1 by Barnett pro­claimed
on the basis of a questionable experiment that “thus the moving force line theory is disproved,”
Kennard’s conclusion was apparently wrong, since his, Kennard’s as well as Barnett’s results can
be easily explained equally well on both the M and N hypotheses, as the mathematical model derived in this paper clearly shows. However, Kennard’s conclusion was apparently accepted by most physicists as correct.
From the point of view of the mathematical model of the generalized homopolar generator presented in this paper, it is clear that no charge could be found in Barnett’s experiment.
48
Namely, using the notation of this paper, Barnett’s experiment involved Z1=1/sC1, ω1= 0, Z2 =
r2 (very small), and ω2 = 0. Of course, the impe­dances can be approximated even better by including the inductive components, but that is absolutely unneces­sary. With those values, Eqs.
(25) and (28) yield zero for the EMF, regardless of whether the magnet rotates or not. The capacitor cannot be charged, which Barnett found to be true.1
Kennard,3 rotating only a coaxial iron bar inside an electromagnet at rest with the stationary electrometer directly connected to a stationary coaxial capacitor, found no charge and
concluded, obviously rather hasti­ly, that “thus the moving force line theory is dis­proved.” Kennard’s measuring procedure is obviously identical to Barnett’s procedure, but the elimination
of the short circuiting simplifies the measurements and provides the direct results.
Without going into the rather heated debate between Barnett and Kennard of how many
lines the rotating iron bar carried with the rotation, Kennard’s results showed no charge. With
the notation used in this paper, Kennard’s experiment described in Ref. 3 involved Z1 = 1/sC1 ,
ω1 = 0, and Z2 = l/sC2 , where C2 is the capaci­tance of the electrometer and ω2 = 0 , neglecting leakage resistances of the coaxial capacitor C1 and the electrom­eter, i. e., the capacitor C2 ,
as well as the small series resistances and the small self-inductances. The induced EMF is zero
according to Eqs. (25) and (28), regard­less of whether the magnet or the iron bar rotated or
not, carrying or not carrying the magnetic field with the rotation. This confirms Kennard’s results reported in Ref. 3.
In order to answer the criticism to his somewhat questionable experiment, Kennard4 performed a more sophisticated experiment with the possibilities of rotat­ing the electromagnet
and the cylindrical coaxial ca­pacitor either together or separately, while the elec­trometer was
always stationary and directly connected to that coaxial capacitor by two brushes (cf. Fig. 2).
Kennard4 found no charge (measured zero voltage by the electrometer) when the electromagnet was rotated, while the cylindrical coaxial capacitor was at rest to­gether with the electrometer, which completely con­firmed Barnett’s results.1 Kennard4 found charge (mea­sured voltage by the electrometer) when the electromag­net and the coaxial capacitor rotated together,
while the stationary electrometer was connected to the rotat­ing coaxial capacitor by two appropriate brushes, of which Kennard complained as causing troubles. This, Kennard’s result,
shows that a coaxial capacitor rotat­ing in the magnetic field is a voltage generator as far as the
stationary observer is concerned. Kennard,4 on the basis of all those unquestionable experimental re­sults, again concluded, but apparently erroneously, that “thus the moving force line
theory is disproved. “ Kennard in Ref.... 4 and elsewhere continued the rather heated controversy with Barnett.
Kennard’s experiment described in Ref. 4 involved, from the point of view of the generalized homopolar generator model Z1 = 1/sC1 , ω1 = 0 in the first phase of the experiment and
ω1 equal to a finite value during the second phase of the experiment, Z2 = 1/sC2 , where C2 is
the capacitance of the stationary electrometer and ω2 = 0 , neglecting again the large leakage
resis­tances (assuming that they are infinitely large), the small series resistances, and the small
self-inductances.
49
With ω 1 ≠ 0 and the switch closed at t = 0 , Eq. (25) or (28) yields for the EMF
u = ω1(WB – WA) h(t) , whose Laplace transform is U = (1/s) ∙ ω1(WB – WA) . Note that by definition 1/s = L{h(t)} , where h(t) is the unit step function. Equation (13) yields for the Laplace
transform of the current through that circuit
.
(30)
By definition, the Laplace transform V2 = V2(s) of the voltage across impedance Z2 is
V2 = Z2I = I/sC2 , whose inverse Laplace transform, utilizing Eq. (30), is
.
(31)
This is precisely the voltage measured by Kennard us­ing the stationary electrometer during
the second phase of the experiment. The rise time is idealized, since the resistances have been
neglected.
Actually, assuming that RA and RB are the radii of the armatures of the coaxial capacitor
with RB > RA , and that the magnetic induction density field B within the solenoid used by Kennard,3,4 Barnett,1 and also by Pegram5 is approximately uniform, then the expression (31) can be
written approximately
.
(32)
This is the result quoted by Kennard4 which is fully in accordance with the mathematical
model of the general­ized homopolar generator, which is presented in this paper.
During the first phase of the experiment, ω1 = 0, the EMF is zero, and the capacitor cannot be charged.
All those results have been found by Kennard and are reported in Ref. 4. But those results do not disprove the M hypothesis as Kennard has claimed, since they are fully in agreement with the mathematical model of the homopolar generator presented in this paper. Barnett1 was right that the Barnett-Kennard experiment could not and did not prove or disprove
either the M or the N hypothesis. However, Barnett was unable to convince his fellow physicists that he was correct, since Barnett himself believed for a long time that the basic premise behind that Barnett-Kennard experiment was correct. Barnett sensed the error at the last
mo­ment through his imagination, which is more important than knowledge, to quote Einstein. Thus, the Barnett-Kennard experiment apparently proved nothing at all, although most
physicists, tired of that old and stubborn puzzle, and also excited at that time by the fresh and
revolutionary ideas of Bohr, accepted Kennard’s con­clusion as correct, a view which is held
even nowadays by most physicists, notwithstanding some obvious diffi­culties and inconsistencies (see Ref. 8, p. 363).
50
It is strange that Kennard did not realize that in his experimental approach the mainly resistive branches of the classical homopolar generator were actually re­placed by chiefly capacitive branches, and that the mag­netic field of the solenoid of the electromagnet was not confined
to the interior space of the solenoid but was also present in the outside space. And the fundamental error appears to be the misconception that an uncharged capacitor during the transient
period of the experiment is an open circuit, which led to the apparently erroneous application
of Faraday’s law of electromagnetic induc­tion without closing the loop of integration during the
calculation of the EMF, and it takes the EMF to charge a capacitor.
It must be mentioned here that apparently a number of other renowned scientists, beside
Barnett himself, such as Poincare, Abraham, Hertz, etc., also held the view that the measurements
with the open circuit of the homopolar generator in the Barnett-Kennard experi­ment could not
resolve the puzzle whether the M or the N hypothesis is correct. However, unfortunately, nei­
ther Barnett nor anybody else so far has put forward strong and correct arguments against Kennard’s er­roneous conclusion that “thus the moving force line theory is disproved.” Barnett came
most closely to the truth, although he actually sensed the error at the last moment.
As a matter of fact, the confusion about this problem in the published literature is so great
that even some experimental results are claimed to have been mea­sured, although those experimental results obviously violate the Faraday induction law, even for the conven­tionally closed circuit (cf. Ref. 4, p. 180). In that re­spect it must be mentioned that Pegram5 reported ex­perimental
results which violate Faraday’s law. Name­ly, Pegram5 rotated the coaxial capacitor inside an
electromagnet together with the rod, which short cir­cuited the capacitor for a moment during
the rotation, in order to charge that capacitor. Clearly, since ω1 = ω2 for such a case, the EMF
is identically equal to zero according to Eq. (25) or (28), regardless of whether the magnet rotates or not, and the capacitor cannot be charged. Pegram, however, reported to have measured
the charge within that capacitor, which ap­pears to be the violation of the Faraday induction law
and, as a consequence, the violation of the principle of the conservation of energy. Pegram’s result is definite­ly at variance with the mathematical model of the gene­ralized homopolar generator presented in this paper.
However, at the time of his experiment, Pegram already believed that the M hypothesis
was disproved by Kennard, and so Pegram expected that result, which he claimed to have measured. Pegram’s description in Ref. 5 casts doubts over his results. One possible ex­planation of
Pegram’s result to have found charge in that case is that Pegram has actually stopped the shortcircuiting rod for a very short period of time during the opening process, Pegram used some
stationary rod to push open the short-circuiting rotating rod instead of securing a strict radial
opening of that short-circuiting rod without any circular component of velocity added or substracted from the actual rotating velocity of that rod. Indeed, such a short stopping of the shortcircuiting rotating rod is almost inevitable unless very special pre­cautions are made. Even if the
time interval of stop­ping the short-circuiting rod was of the order of 1 nsec. of which possibility Pegram was apparently unaware and the measurements of which were impossible during Pegram’s lifetime, but such a short stopping must be considered as inevitable due to the inertia,
then in view of the estimated capacitance of the order of 100 pF and the estimated resistance of
51
the short-circuiting rod of the order of 10 mΩ thus yielding the time constant of the circuit of
the order of 1 psec, the capacitor must have been charged almost inexorably. Thus, Pegram’s result violates the Faraday induction law, and it ap­pears to favor the N hypothesis, which Pegram
believed to have been proved by Kennard. Thus, it is only Pegram who could have claimed to
have disproved the M hypothesis, which he never did. Pegram also report­ed to have performed
the experiment with the stationary capacitor and the stationary short-circuiting rod while the
magnet rotated, confirming Barnett’s result, Pegram’s report about that part of the experiment
is contained in a single sentence, without any numerical values at all, which is strange and suggests that Pegram was not very careful. As a matter of fact, Pegram’s paper appeared when the
physicists of that bygone era no longer cared about that problem, since the problem has been already considered as “solved” and “closed.” It is very probable that nobody read Pegram’s paper5
very carefully at that time.
Actually, Pegram’s result not only contradicts the mathematical model presented in this paper, but it also contradicts Kennard’s unquestionable result,4 as well as that of many other physicists (cf. Ref. 2, p. 325), i.e., that a rotating uncharged coaxial capaci­tor in the magnetic field is
observed by the stationary observer as a source of the voltage which, using the same notations
and approximations as for the expres­sion (32), can be written in the form
.
(33)
Thus, in view of the generalized Thevenin theorem, the rotating coaxial uncharged capacitor in the magnetic field is equivalent to the voltage generator, whose volt­age u1 is given by Eq.
(33) and is measured across the stationary brushes A and B, i.e., L{u1}, in series with the capacitive internal impedance 1/sC1 . The Laplace transform notation is used.
On the other hand, the rotating metallic rod is fully equivalent to and can be replaced by a
rotating disk, provided the attachments, i.e., brush points, remain the same. Assuming RA and
RB as the radii of the brushes A and B, the voltage of the disk (Faraday disk) rotating with the
constant angular velocity ω2 in the approximately uniform magnetic field B , as used for Eqs.
(32) and (33), is given by
.
(34)
Thus, in view of the generalized Thevenin theorem, the rotating disk, or rod to that matter,
is equivalent to the voltage generator, whose voltage u2 is given by Eq. (34), which is measured
across the stationary brushes A and B, in series with the small internal resistance r2 .
Note that the polarities of u1 and u2 are such that they are in opposition. When ω1 ω2 ,
i.e., when the capaci­tor and the disk or rod are rotated together, while the brushes are stationary, or even removed as unneces­sary, as Pegram did in his experiment, the voltages u1
and u2 cancel each other, and no electrical current observed by the stationary observer can
flow, which may result in the separation of the mobile charges. Consequently, the coaxial capacitor could not have been charged, as reported by Pegram, i.e., Pegram’s re­sult appears to
have been an error.
52
Actually, Pegram’s result implies that either the coaxial rotating capacitor or the rotating rod,
i. e., disk, is not the source of the potential difference, i. e., voltage. But that contradicts experimental results by many physicists, including Kennard.4 A rotating co­axial capacitor in the axial magnetic field is definitely a voltage generator with the finite capacity for the de­livery of the
electrical charge, due to its capacitive internal impedance. The Faraday disk or the rotating rod
in the axial magnetic field is also a voltage genera­tor. Those two voltages in Pegram’s experiment
cancel each other according to the mathematical model pre­sented in this paper. Pegram’s claimed
result implies that it is irrelevant whether one of those two structures is rotating or not. Carried
even further, that would im­ply that one structure could rotate in the opposite direc­tion or with
any arbitrary angular velocity without changing the result, i. e., the result remains the same under arbitrarily large numbers of distinctly different physical conditions. That appears to be absurd, al­though it is logically possible but very improbable. Thus, Pegram’s result appears clearly
to be an error, which violates the Faraday law of electromagnetic induction.
CONCLUSION
It has been shown that the admissible circular trajec­tory of a test point electrical charge in the
equatorial plane of the fixed point magnetic dipole can be inter­preted by the stationary observer as the revolution of the test point charge around another electrical point charge. On the other hand, for the observer attached to the test point charge, it appears that the point magnetic dipole is revolving around that observer, and at the same time the magnetic dipole is spinning with
the synchronous angular velocity in the negative direction. This dynamic problem suggests that
a spinning time-invariant point magnetic dipole may be considered as a possible model of the
electrical charge within the frame­work of classical physics, provided the M hypothesis was not
disproved, as claimed by Kennard.3,4 but con­travened unsuccessfully by Barnett.1,2
In view of the potentially high significance of the quantum-mechanical electrical charge
model proposed recently and extensively analyzed by Jehle, 6,7 and in view of the intriguing possibility of the electrical charge model within the framework of classical physics as ex­posed above,
which is basically similar to Jehle’s model, the controversy at the turn of this century con­cerning
the moving or the nonmoving force line theory was reanalyzed.
That analysis reveals the fact that if the mathemati­cal model of the generalized homopolar
generator, de­rived in this paper, is used, then the claim by Kennard that “thus the moving force
line theory is disproved” must be considered as incorrect, since all results by Barnett, Kennard,
and others, except for a single re­sult by Pegram,5 can be explained on either the M or the N hypothesis. Pegram’s result might be argued as dis­proving the M hypothesis, but that result also violates the Faraday law of induction and it appears to be an error. Pegram believed when he made
that experiment that the M hypothesis was already disproved. But the M hypothesis was not disproved as shown, and it ap­pears definitely to be more logical and preferable, un­less the concept of
the field is a pure abstraction, de­void of any physical meaning.
Further experimental research with all kinds of spin­ning magnets, including the repetition of Pegram’s ex­periment, appears to be very interesting from the point of view of search
53
for truth, with a view towards an in­triguing model of the electrical charge. The controversy at
the turn of this century has apparently not resolved the old but stubborn puzzle of the moving or the non-moving force line theory which, to the best of imagina­tion and knowledge of
this author, may prove, after all, to be an antinomy or the breakdown of the classical electromagnetic field theory.
References
1.S.J. Barnett, Phys. Rev. 33, 323 (1912).
2.S.J. Barnett, Phys. Rev. 2, 323 (1913).
3. E.H. Kennard, Philos. Mag. 23, 937(1912).
4. E.H. Kennard, Philos. Mag. 33, 179(1917).
5. G. B. Pegram, Phys. Rev. 10, 591 (1917).
6. H. Jehle, Phys. Rev. D 3, 306 (1971).
7. H. Jehle, Phys. Rev. D 6, 441 (1972).
8.A. Sommerfeld, Electrodynamics (Academic, New York, 1952).
9.S. Tolver Preston, Philos. Mag. 31, 100 (1891).
10.Tracy D. Waring, Trans. Am. Inst. Electr. Eng. 27, 1366 (1908).
54